Interphase Drag Coefficients in Gas Solid Flows

Transcription

1 R&D NOTES Interphase Drag Coeffcents n Gas Sold Flows D. Kandha, J. J. Derksen, and H. E. A. Van den Akker Kramers Laboratorum voor Fyssche Technologe, Faculty of Appled Scences, Delft Unversty of Technology, Prns Bernhardlaan BW Delft, The Netherlands Gas sold fludzed-bed reactors are commonly used n the process ndustres due to ther hgh operatonal effcency regardng, for nstance, solds mxng and heat and mass transfer Ž Geldart, It s well known that hydrodynamcs plays a crucal role n the dynamc behavor of fludzed beds Ž Geldart, 1986; Fan and Zhu, Currently, computatonal flud dynamcs methods are wdely used to obtan a better understandng of the complex behavor due to gas sold Ž hydrodynamc. and sold sold Ž collsonal. nteractons and, of course, how ths emergent behavor affects the operaton of fludzed-bed reactors Žsee, for example, Goldschmdt et al., Numercal smulatons of flud dynamcs n such systems are generally based on the so-called two-flud models ŽFan and Zhu, In these models, both phases are consdered as nterpenetratng contnua. Mass, momentum, and Žn some cases. energy balances are derved usng volume and tme or ensemble averagng technques. For example, the momentum equatons for the gas phase n the case of sothermal flow follow the relaton Ž gu. qž guu. sy py Ž uy. y Ž. t q g g Ž. 1 Here u and are the velocty of the gas and sold phase, respectvely; s the gas volume fracton; g s the densty of the gase; p s the pressure; s the stress tensor; g s the gravtatonal acceleraton, and s the nterphase drag coeffcent, whch s the prmary subject of ths study. Ths term s a closure relaton for the drag force exerted on the sold partcles by the gas phase, as a functon of the sold volume fracton, s1y, and the partcle Reynolds number, Res Ž uy 2 a. r Žwth a the radus of the sphercal sold partcles and the knematc vscosty of the gas phase.. Correspondence concernng ths artcle should be addressed to D. Kandha. In many cases, the well-known Wen and Yu Ž and the Ergun Ž equatons are used to descrbe nterphase drag coeffcents. The Ergun correlaton expressed n terms of the nondmensonal drag force actng on a sngle partcle, F, s gven by 0.18 Re F s8.33 q Ž Ž 1y. 2 Ž 1y. Here F s made dmensonless by the gas-volume fracton,, and the Stokes drag actng on a sngle sphere n an nfnte medum, F s6 a uy. s g The Wen and Yu Ž correlaton follows the relaton y4.65 Ž.Ž. Ž. F s 1q0.15Re 1y 3 Notce that F s related to through the expresson F s Ž 2a. 2 rž 18 2 Ž 1y.. g. The Wen and Yu correlaton, a refnement of the Rchardson and Zak equaton Ž 1954., s based on partcle fludzaton experments performed n a wde range of sold-volume fractons and Reynolds numbers, and 0.01 Re 5,000, respectvely. The Ergun equaton, on the other hand, s derved from pressure-drop measurements n closed-packed fxed beds. Furthermore, although the latter s based on systems wth partcles on fxed postons, t can be appled to dynamc systems, as well, f the densty rato between the two phases Ž as generally s the case n gas sold flow. s large Ž Koch, In numercal smulatons of gas sold fludzed-beds, t s stll unclear whch correlaton should be used for descrbng the nterphase drag coeffcents. The two wdely used models are the Wen and Yu equaton and a hybrd model, suggested by Gdaspow Ž 1994., wth Wen and Yu for 0.2 and Ergun otherwse. In a recent study, t has been shown by van Wachem et al. Ž that the exact choce of the closure relaton can have a sgnfcant nfluence on, for example, the smulated bubble shapes n fludzed beds Aprl 2003 Vol. 49, No. 4 AIChE Journal

2 Recently, Hll et al. Ž performed a rgorous study concernng the dependence of the drag force on Re and for a wde range of sold-volume fractons. By usng a lattce Boltzmann method, the drag force actng on fxed Ž ds. ordered monodsperse bead packngs n perodc domans was computed. For dense systems Ž G0.5., the smulated drag curve s n good agreement wth the Ergun equaton, n contrast to that n the less dense regme. However, a connecton wth the Wen and Yu equatons was not reported, leavng the followng queston unresolved: Whch should one dfferentate between the dlute and dense regmes? In ths artcle, we revst the drag-force closures problem and focus on the queston: Is the hybrd model ndeed vald, or should we nstead dscrmnate between a dlute, dense, and an ntermedate regme? For ths we perform lattce Boltzmann Ž Chen and Doolen, smulatons of flud flow n fxed monodsperse dsordered perodc sphere packngs n the range 1 Re 50 and , and compare our results wth that obtaned by Hll et al. Ž and wth the Wen and Yu and the Ergun correlatons. Smulaton Method In smulatons, we consder flud flow through dsordered arrays of spheres n a perodc box. For the computaton of the flow felds and the drag force, we use the lattce Boltzmann method Ž LBM.. In the past decade, LBMs have proven to be versatle tools n smulatng a wde varety of applcatons, rangng from creepng flow n porous meda ŽKandha et al., 2002a,b. to turbulent flows n strred-tank reactors Ž Derksen, These methods orgnated from the lattce gas automata Ž LGA., whch are dscrete models for the smulaton of transport phenomena. In LGA, fcttous partcles move synchronously along the bonds of a regular lattce and nteract locally accordng to a gven set of rules subject to conservaton of mass and momentum. Due to ths nherent spatal and temporal localty, these methods are well suted for parallel computng. In contrast to LGA, n LBM a densty of partcles s beng tracked rather than a sngle one, and there s relatvely more freedom n the formulaton of the collson operator Ž Chen and Doolen, The smplest collson model s the Bhatnagar Gross Krook Ž BGK. scheme wth a sngle tme relaxaton to the local equlbrum dstrbuton. The correspondng lattce BGK method ŽChen and Doolen, s gven by 1 Ž0. fž rq c,tq1. s fž r,t. q f Ž r,t. y fž r,t. Ž 4. where c s the th lnk; f Ž r,t. s the densty of partcles mov- ng n the c-drecton; s the BGK relaxaton parameter; and f 0 Ž r,t. s the equlbrum dstrbuton functon toward whch the partcle populatons are relaxed. A common choce for f 0 Ž r,t. s, f st 1q Ž c u. q Ž c u. y u Ž c 2c 2c s s s where t s a weght-factor, whch depends on the length of the vector c ; c s the speed of sound; and s the densty. s The densty and the velocty are obtaned from moments of the dscrete velocty dstrbuton f Ž r,t. N Ý f s 0 Ž r,t. c N Ž r,t. s Ý f Ž r,t. and už r,t. s Ž 6. Ž r,t. s 0 wth N denotng the number of lnks per lattce pont. The knematc vscosty n lattce unts Ž l.u.. s gven by sž y 1r2. r3. For our current purpose, we use the so-called DQ 3 19 lattce BGK model Ž Chen and Doolen, In the smulatons, the flow s drven by a body force, that s, at each tme step a fxed amount of momentum s added to all lattce ponts. The sold flud nterface s modeled by usng the well-known bounce-back method, that s, partcles enterng a sold node are reflected to the flud wth reversed velocty. Steady state n the smulaton s acheved when the total body force actng on the flud s equal to the drag force exerted on the sold matrx. The Reynolds number s now vared by adjustng the body force. Results and Dscusson Generally, the computatonal grd n LB smulatons s unform and Cartesan. Extenson of the method to rregular grds s avalable, although ts applcaton s stll restrcted to sngle-partcle systems due to lmted computatonal resources Ž Rohde et al., As a consequence, the dscrete representaton of a sphere n dsordered arrays s starcased. The starcasng of the surface, the way sold flud boundares are mposed and the actual accuracy of the method, are reflected n the dscretzaton error Ž Kandha et al., From prelmnary fnte-sze studes, we conclude that, for the low sold fractons, a sphere dscretzaton wth a dameter of 10 lattce ponts yelds satsfactory results, that s, dscretzaton errors are then around 10%. For the denser systems, known to be much more senstve to numercal errors due to the lmted number of channels through whch the flud can percolate, we have to dscretze the spheres wth a dameter of some 20 lattce ponts to obtan smlar accuraces Žsee Fgure 1.. In LBM schemes, yet another artfact exsts, namely, a dependence on the flud vscosty of the radus of partcle that s actually beng smulated Žthe so-called hydrodynamc radus.. One way to correct for ths effect s by applyng the followng procedure Ž Ladd, 1994.: Frst, the hydrodynamc radus correspondng to a certan geometrcal radus and knematc vscosty s computed by usng the analytcal soluton of creepng flow through a perodc array of spheres Ž n the dlute lmt. ŽHasmoto, Next, the many-partcle computatons are calbrated by takng the hydrodynamc radus as the true radus of the partcles. The correspondng hydrodynamc rad of spheres n a flud flow wth knematc vscosty of Ž n l.u.. and dscretzed usng 10 and 20 lattce ponts, are 10.4 and 20.4, respectvely Ž n l.u... In Fgure 2, we show the results obtaned for the drag force as a functon of the Reynolds number for f0.3 Ž after calbraton.. It s clear, that after havng appled the AIChE Journal Aprl 2003 Vol. 49, No

3 Fgure 1. Fnte-sze smulatons: relatve error ( n %) based on the 2a=30 case, n the drag force computed on dfferent grd resolutons. In all smulatons, the postons of the partcles are the same. The volume fracton of the system s 0.62 Žworst-case scenaro.. The relaxaton parameter s1.0. calbraton procedure, the results obtaned by usng rad of 10 Ž s0.29. and 20 lattce ponts Ž s0.274., respectvely, match satsfactorly well. Some dfferences may be expected due to a msmatch n the exact value of. Apart from the ssues related to resoluton senstvty, the computed drag forces suffer from statstcal fluctuatons due to heterogenetes assocated wth the arrangements of the spheres. These effects can be reduced by averagng the results obtaned by a seres of smulatons wth dfferent geometrcal confguratons. The mean drag-force s defned as the ensemble average of the drag forces obtaned by nc nde- pendent smulatons Ž. Analogous to the work of Hll et al. 2001, the uncertanty n the estmated mean s computed from s( varž F. F n c y1 wth varž F. s Ž F y F. 2 denotng the varance of F Ž see Hll et al., 2001, and references theren.. The geometrcal confguraton of the spheres s generated by the method descrbed n ŽKandha et al. Ž 2002a.. Note that n the extreme case of closed packed systems, the structure of the medum used n the computatons mght be dfferent as compared to that of the packed beds used n the expermental measurements. It should be noted that n the smulatons consdered here, the partcle postons are held fxed. As dscussed n the frst secton, n two-flud models, the macroscopc equatons are derved based on some averagng prncple. The presence of a certan amount of sold phase s then modeled by the sold volume fracton parameter, s. Furthermore, there s no d- rect characterzaton of the geometrcal stucture of the sold partcles. Therefore, drag relatons obtaned from ensemble averagng of the systems consdered here are useful as closure relatons n the two-flud models. Apart from that, n gas sold flows, the response tme of the partcle dynamcs Ž ŽŽ.Ž 2 due to dsturbances n the flow s r d r18.. p p f p, s very large on the order of 1,000 compared to the tmescale related to hydrodynamc dsturbances n flud Ž c s d ru. p s. It s, therefore, expected that the ensemble-averaged behavor of systems wth fxed partcles may be smlar to that of homogeneous fludzed systems n the equlbrum state. In Fgure 3, we show the average slp velocty as a functon of 1 ² F: s n Ý F n c s1 Fgure 2. Fnte-sze smulatons: calbraton of the smulaton results by the hydrodynamc radus. The volume fracton of the packed bed s 0.29 Ž for as10. and Ž for as 20.. The relaxaton parameter s 0.7. Fgure 3. Transent behavor of the drag force for a system of freely movng partcles compared to the average drag force obtaned for a fxed array of partcles; the dmenson of the smulaton doman s 8 a 8 a 8 a, =0.27, a=10, Ref 10, and p/ f=1, Aprl 2003 Vol. 49, No. 4 AIChE Journal

4 Fgure 4. Drag force as a functon of Re for a dsordered perodc array of spheres wth varyng sold volume fractons =0.1 to 0.5. Theexactvaluesofthevolumefractonsare:fromŽ. a tož. f s 0.10, 0.19, 0.27, 0.36, 0.43, and The Ergun and Wen and Yu equatons are ncluded n all graphs. Furthermore the results obtaned by Hll are also shown for the volume-fractons wth a close match. The dmensons of the smulaton doman are 8a 8 a 8 a, wtha the radus of the spheres n lattce unts Ž l.u.., as10 for 0.3 and as 20 for 0.3. The number of spheres s vared from 16 to 64. AIChE Journal Aprl 2003 Vol. 49, No

5 the tme of systems of freely movng partcles Žwth s0.27, as10, Ref10, and r s1,000. p f and compare that wth the averaged ensemble slp velocty of systems wth partcle postons fxed. It s evdent that the average slp velocty of the dynamc system s close to that of the fxed system, when we take nto account varatons due to the uncertanty n the estmated mean. Smlar results have been obtaned for other volume fractons Ž f0.1 and f0.2, respectvely.. Notce that the structure of fxed and sedmentng systems mght be dfferent, especally n the case of lqud sold suspensons. The eventual smulatons were carred out for perodc boxes wth dmenson of 8a 8a 8a, the correspondng grd dmensons beng Ž for 0.3. and Ž for 0.3., respectvely. The memory requrements for these smulatons were at most 600 Mb, and the number of tme steps to reach steady state was vared from 10,000 to 2,000 tme steps for ncreasng sold-volume fracton. We now return to our man queston, whether the Wen and Yu and the Ergun correlatons do capture the complete range of sold-volume fractons. Fgure 4 shows, the results for the dmensonless drag force as a functon of the Reynolds number for ncreasng sold-volume fracton. The correspondng curves of the Wen and Yu and the Ergun correlatons are ncluded n all plots. Moreover, n Fgure 4a and 4b, we also have ncluded the results obtaned by Hll et al. Ž These are the only two cases n whch there s a close match n the sold-volume fractons consdered n both studes. We clearly see that there s a good agreement between both smulatons. Ths pont s further confrmed n Fgures 5 and 6. Fgures 5 and 6 show the slope of F vs. Re as a functon of, and the functonal dependence of F on as a functon of the vod fracton, respectvely. As dscussed earler, an eventual transton regon s expected to occur n the range of sold-volume fractons be- Fgure 5. Slope of F vs. Re as a functon of ; the sold lne s the ft expresson obtaned by Hll et al. ( 2001 ). Fgure 6. Functonal dependence of F on as a functon of the vod fracton. The sold lne s the Wen and Yu Ž correlaton. tween 0.2 and 0.4. Therefore, we spend most of our computatonal effort n ths specfc range: the mean drag force and the uncertanty n the estmated mean were computed usng the method descrbed earler wth ncs10. For the other vol- ume fractons, we restrcted the computatons to a sngle geometrcal confguraton, n order to reduce the total computaton tme. Before dscussng the fnal results, we stll would lke to address a subtle pont. As mentoned n the prevous secton, the flow n the smulatons was drven by a body force. Consequently, the average velocty could not be predcted n advance, but was dependent on the geometrcal confguraton. For each geometrcal confguraton, however, we found that a lnear relaton fts all the smulated ponts very well Ž wth errors n the fttng parameters F0.5%.. Therefore, we used the ftted expressons to estmate the drag force for specfc values of Re and then computed the mean based on these values. The frst observaton n Fgure 4 s that our results are ndeed n good agreement wth the Wen and Yu correlaton for the low sold-volume fractons and the Ergun correlaton for the hgh sold-volume fractons. The nterestng feature s that the Wen and Yu correlaton shows a better match wth the smulaton results, even for f0.3. The Ergun equaton s found to be vald for f0.5. It s evdent that for 0.3F F0.43 both the Wen and Yu and the Ergun correlaton show a dscrepancy wth the computed values, whch suggests that there s ndeed an ntermedate regme. If uncertantes on the order of 10% are acceptable, the ft-correlaton obtaned by Hll et al., and found to be consstent wth our results, can be used as a closure relaton for 0 Re 100. However, to provde an even more accurate correlaton for the ntermedate regme, more detaled smulatons based on fner grd resolutons or by usng more sophstcated methods for mposng no-slp boundary condtons n lattce Boltzmann smulatons Ž. Rohde et al., 2002, are stll necessary Aprl 2003 Vol. 49, No. 4 AIChE Journal

6 Acknowledgments The work s part of the research program of the Dutch Research School on Process Technology, Onderzoekschool Proces Technologe Ž OSPT.. It s fnancally supported by Akzo-Nobel, DSM and Shell Research. We thank Prof. Dr. Ir. J. A. M. Kupers, ŽUn- verstet Twente. for many useful dscussons. Lterature Cted Chen, S., and G. D. Doolen, Lattce Boltzmann Method for Flud Flows, Annu. Re. Flud Mech., 30, 329 Ž Derksen, J. J., and H. E. A. Van den Akker, Large Eddy Smulatons on the Flow Drven n a Rushton Turbne, AIChE J., 45, 209 Ž Ergun, S., Flud Flow Through Packed Columns, Chem. Eng. Prog., 48, 245 Ž Fan L. S., and C. Zhu, Prncples of Gas-Sold Flows, Cambrdge Unv. Press, Cambrdge Ž Geldart, D., Gas Fludzaton Technology, Wley, New York Ž Gdaspow, D., Multphase Flow and Fludzaton, Academc Press, San Dego Ž Goldschmdt, M. J. V., B. P. B. Hoomans, and J. A. M. Kupers, Recent Progress Towards Hydrodynamc Modelng of Dense Gas-Partcle Flows, Recent Res. De. Chem. Eng., 4, 273 Ž Hasmoto, H., On the Perodc Fundamental Solutons of the Stokes Equatons and Ther Applcaton to Vscous Flow Past a Cubc Array of Spheres, J. Flud Mech., 5, 317 Ž Hll, R. J., D. L. Koch, and A. J. C. Ladd, Moderate-Reynolds- Number Flows n Ordered and Random Arrays of Spheres, J. Flud Mech., 448, 243 Ž Kandha, D., A. Koponen, A. Hoekstra, M. Kataja, J. Tmonen, and P. M. A. Sloot, Implementaton Aspects of 3D Lattce-BGK: Boundares, Accuracy and a New Fast Relaxaton Method, J. Comput. Phys., 150, 482 Ž Kandha, D., U. Tallarek, D. Hlushkou, A. G. Hoekstra, P. M. A. Sloot, and H. van As, Numercal Smulaton and Measurement of Lqud Holdup n Bporous Meda Contanng Dscrete Stagnant Zones, Phlos. Trans. R. Soc. London A, 360, 521 Ž 2002a.. Kandha, D., D. Hlushkou, A. G. Hoekstra, P. M. A. Sloot, H. van As, and U. Tallarek, Influence of Stagnant Zones on Transent and Asymptotc Dsperson n Macroscopcally Homogeneous Porous Meda, Phys. Re. Lett., 88, Ž 2002b.. Koch, D. L., Knetc Theory for a Monodsperse Gas-Sold Suspenson, Phys. Fluds A, 2, 1711 Ž Ladd, A. J. C., Numercal Smulatons of Partculate Suspensons Va a Dscretzed Boltzmann Equaton. Part 2. Numercal Results, J. Flud Mech., 271, 311 Ž Rchardson, J. F., and W. N. Zak, Sedmentaton and Fludzaton: Part I, Trans. Inst. Chem. Eng., 32, 35 Ž Rohde, M., J. J. Derksen, and H. E. A. Van den Akker, Volumetrc Method for Calculatng the Flow Around Movng Objects n Lattce-Boltzmann Schemes, Phys. Re. E, 65, Ž Van Wachem, B. G. M., J. C. Schouten, C. M. van den Bleek, R. Krshna, and J. L. Snclar, Comparatve Analyss of CFD Models of Dense Gas-Sold Systems, AIChE J., 47, 1035 Ž Wen, Y. C., and Y. H. Yu, Mechancs of Fludzaton, Chem. Eng. Prog. Symp. Ser., 62, 100 Ž Manuscrpt rece ed Feb. 14, 2002, and re son rece ed Sept. 18, AIChE Journal Aprl 2003 Vol. 49, No